Generalized Ricci flow I: Local existence and uniqueness

TL;DR

This paper establishes the short-time existence and uniqueness of a generalized Ricci flow with gradient form on compact manifolds, analyzing its properties and evolution equations for curvatures.

Contribution

It introduces a generalized Ricci flow with gradient form, proves its short-time existence and uniqueness, and derives curvature evolution equations.

Findings

Flow is strictly and uniformly parabolic

Unique short-time smooth solutions exist

Derived evolution equations for curvatures

Abstract

In this paper we investigate a kind of generalized Ricci flow which possesses a gradient form. We study the monotonicity of the given function under the generalized Ricci flow and prove that the related system of partial differential equations are strictly and uniformly parabolic. Based on this, we show that the generalized Ricci flow defined on a $n$-dimensional compact Riemannian manifold admits a unique short-time smooth solution. Moreover, we also derive the evolution equations for the curvatures, which play an important role in our future study.